{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T14:58:02Z","timestamp":1778511482927,"version":"3.51.4"},"reference-count":41,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,20]],"date-time":"2021-10-20T00:00:00Z","timestamp":1634688000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Key R&amp;D Program of China","award":["2017YFB0203604"],"award-info":[{"award-number":["2017YFB0203604"]}]},{"name":"the National Natural Science Foundation of China","award":["11772077"],"award-info":[{"award-number":["11772077"]}]},{"name":"the LiaoNing Revitalization Talents Program","award":["XLYC1807187"],"award-info":[{"award-number":["XLYC1807187"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, the topology optimization of continuum structures with design-dependent loads is studied with a gradient-free topology optimization method in combination with adaptive body-fitted finite element mesh. The material-field series-expansion (MFSE) model represents the structural topology using a bounded material field with specified spatial correlation and provides a crisp structural boundary description. This feature makes it convenient to identify the loading surface for the application of the design-dependent boundary loads and to generate a body-fitted mesh for structural analysis. Using the dimension reduction technique, the number of design variables is significantly decreased, which enables the use of an efficient Kriging-based algorithm to solve the topology optimization problem. The effectiveness of the proposed method is demonstrated using several numerical examples, among which a design problem with geometry and contact nonlinearity is included.<\/jats:p>","DOI":"10.3390\/sym13111976","type":"journal-article","created":{"date-parts":[[2021,10,20]],"date-time":"2021-10-20T22:07:04Z","timestamp":1634767624000},"page":"1976","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["A Gradient-Free Topology Optimization Strategy for Continuum Structures with Design-Dependent Boundary Loads"],"prefix":"10.3390","volume":"13","author":[{"given":"Junjie","family":"Zhan","sequence":"first","affiliation":[{"name":"School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, China"},{"name":"School of Architecture and Art Design, Hebei University of Technology, Tianjin 300130, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jing","family":"Li","sequence":"additional","affiliation":[{"name":"China Academy of Launch Vehicle Technology, Beijing 100076, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pai","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5017-0639","authenticated-orcid":false,"given":"Yangjun","family":"Luo","sequence":"additional","affiliation":[{"name":"School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/0045-7825(88)90086-2","article-title":"Generating optimal topologies in structural design using a homogenization method","volume":"71","author":"Kikuchi","year":"1988","journal-title":"Comput. 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